On compressible Korteweg fluid-like materials

Heida, Martin; Málek, Josef

doi:10.1016/j.ijengsci.2010.06.031

Cited by 72 publications

(56 citation statements)

References 15 publications

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“…Making different choices for the specific internal energy, different equations like those governing the Stefan problem, the Cahn-Hilliard equation, etc., can be derived. In the case of a single constituent, assuming the dependence of the specific internal energy on the concentration and its gradient, and requiring that the rate of entropy production be maximized leads naturally to the model developed by Korteweg [20], as has been shown earlier by Heida and Málek [13].…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we shall be primarily interested in Cahn-Hilliard type of models, we shall primarily make choices for the Helmholtz potential and the rate of entropy production that lead to such models. However, it is possible to show that the framework also leads to several other types of models such as those due to Korteweg [20] to describe capillarity as shown in [13]. In order to develop all these models (generalizations of Korteweg, Cahn-Hillard, Stefan, Cahn-Allen equations, etc.…”

Section: Introductionmentioning

confidence: 99%

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On the development and generalizations of Cahn–Hilliard equations within a thermodynamic framework

Heida

Málek

Rajagopal

2011

Z. Angew. Math. Phys.

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We provide a thermodynamic basis for the development of models that are usually referred to as "phase-field models" for compressible, incompressible, and quasi-incompressible fluids. Using the theory of mixtures as a starting point, we develop a framework within which we can derive "phase-field models" both for mixtures of two constituents and for mixtures of arbitrarily many fluids. In order to obtain the constitutive equations, we appeal to the requirement that among all admissible constitutive relations that which is appropriate maximizes the rate of entropy production (see Rajagopal and Srinivasa in Proc R Soc Lond A 460: 2004). The procedure has the advantage that the theory is based on prescribing the constitutive equations for only two scalars: the entropy and the entropy production. Unlike the assumption made in the case of the Navier-Stokes-Fourier fluids, we suppose that the entropy is not only a function of the internal energy and the density but also of gradients of the partial densities or the concentration gradients. The form for the rate of entropy production is the same as that for the Navier-Stokes-Fourier fluid. As observed earlier in Heida and Málek (Int J Eng Sci 48(11):1313-1324, it turns out that the dependence of the rate of entropy production on the thermodynamical fluxes is crucial. The resulting equations are of the Cahn-Hilliard-Navier-Stokes type and can be expressed both in terms of density gradients or concentration gradients. As particular cases, we will obtain the Cahn-Hilliard-Navier-Stokes system as well as the Korteweg equation. Compared to earlier approaches, our methodology has the advantage that it directly takes into account the rate of entropy production and can take into consideration any constitutive assumption for the internal energy (or entropy). Mathematics Subject Classification (2000). Primary 76A02 · 76T30; Secondary 00A71 · 76A99.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the development and generalizations of Cahn–Hilliard equations within a thermodynamic framework

Heida

Málek

Rajagopal

2011

Z. Angew. Math. Phys.

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“…Second and higher gradient models in continuum mechanics are becoming more and more popular whenever it is needed to model boundary layers and are nowadays known as the phase-field models, see the review papers [8,21,43,52,72,85,106,110] and the reference therein.Here we use the latter approach based on the model of second gradient fluid which is also called Korteweg or Cahn-Hilliard fluid. The model relates to the original works of Korteweg [74] and Cahn-Hilliard [11,12], see also [10,42,54,70,93,[101][102][103]. It has to be remarked that second gradient fluids can exert shear stresses (see e.g.…”

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confidence: 96%

Propagation of linear compression waves through plane interfacial layers and mass adsorption in second gradient fluids

2013

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This paper addresses the problem of reflection and transmission of compression waves at the phase transition layer between the vapour and liquid phases of the same fluid. Within the framework of second gradient fluid modeling, we use a non-convex free energy in order to describe the phase transition phenomenon. A stationary solution for the fluid density is found for an infinite domain, and an analytical expression for the phase transition is presented. Then the propagation of linear waves superposed to this stationary solution is discussed, with particular attention to the behaviour in correspondence of the interfacial layer. The reflection and transmission of waves is studied and analized with the aid of numerical simulations, and an interesting phenomenon of mass adsorption at the interface is observed and discussed. (C) 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinhei

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“…Their derivation leads to a Korteweg stress tensor term as commonly seen in Navier-Stokes-Korteweg models. 4,21,25,32,42 Other simpler models have also been introduced, 11,15,47 which neglect certain terms in the above-mentioned quasi-incompressible models. These models seem to be inconsistent with mixture theory and the second law of thermodynamics, and have been studied mostly because of their simpler implied numerical treatment, which is closer to that of the variable-density Navier-Stokes equations 26 and Volume-of-fluid (VOF) method.…”

Section: Lowengrub and Truskinovskymentioning

confidence: 99%

Diffuse-interface two-phase flow models with different densities: A new quasi-incompressible form and a linear energy-stable method

Roudbari

Şimşek

Brummelen

et al. 2018

Math. Models Methods Appl. Sci.

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While various phase-field models have recently appeared for two-phase fluids with different densities, only some are known to be thermodynamically consistent, and practical stable schemes for their numerical simulation are lacking. In this paper, we derive a new form of thermodynamically-consistent quasi-incompressible diffuse-interface Navier-Stokes Cahn-Hilliard model for a two-phase flow of incompressible fluids with different densities. The derivation is based on mixture theory by invoking the second law of thermodynamics and Coleman-Noll procedure. We also demonstrate that our model and some of the existing models are equivalent and we provide a unification between them. In addition, we develop a linear and energy-stable time-integration scheme for the derived model. Such a linearly-implicit scheme is nontrivial, because it has to suitably deal with all nonlinear terms, in particular those involving the density. Our proposed scheme is the first linear method for quasi-incompressible two-phase flows with nonsolenoidal velocity that satisfies discrete energy dissipation independent of the time-step size, provided that the mixture density remains positive. The scheme also preserves mass. Numerical experiments verify the suitability of the scheme for two-phase flow applications with high density ratios using large time steps by considering the coalescence and break-up dynamics of droplets including pinching due to gravity.

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On compressible Korteweg fluid-like materials

Cited by 72 publications

References 15 publications

On the development and generalizations of Cahn–Hilliard equations within a thermodynamic framework

On the development and generalizations of Cahn–Hilliard equations within a thermodynamic framework

Propagation of linear compression waves through plane interfacial layers and mass adsorption in second gradient fluids

Diffuse-interface two-phase flow models with different densities: A new quasi-incompressible form and a linear energy-stable method

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